Why can't quantum field theory be quaternion instead of complex?

So, the definition of QFT in terms of path integrals is that the partition function is:

$$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$

But does it have any meaning if instead of this $U(1)$ quantum mechanics you replace it with $SU(2)$ of unit quaternions:

$$Z[J] \propto \int e^{iS_1[\phi]+jS_2[\phi]+kS_3[\phi]+J.\phi} D[\phi]$$

Obviously there are three actions $S$ instead of one. So is this kind of thing forbidden? Or is it equivalent to something else? (i.e. could all 3 actions be combined into one?) Is there something special about complex numbers? What is the physical principle or mathematical principle that says that we must only consider complex $U(1)$ phases.